The Shapley value of exact assignment games - Springer Link

Int J Game Theory (2007) 35:557–568 DOI 10.1007/s00182-006-0068-8 O R I G I NA L PA P E R

The Shapley value of exact assignment games Maike Hoffmann · Peter Sudhölter

Published online: 2 February 2007 © Springer-Verlag 2007

Abstract We prove that the Shapley value of every two-sided exact assignment game lies in the core of the game.

1 Introduction The Shapley value and the core are widely accepted and frequently applied solutions for cooperative transferable utility games. The Shapley value of a coalitional game is a singleton and it may be interpreted as an a priori evaluation of the game. Each element of the core of a coalitional game is stable in the sense that no coalition can improve upon this element. In view of these interpretations (see Sect. 2 for the formal definitions) it is not surprising that the Shapley value may not select a core element, even if the core is nonempty. Indeed, for asymmetric glove games, the Shapley value does not select the unique element of the core (the unique competitive allocation). Though there exists a convergence theorem (see Shapley and Shubik 1969), our approach is rather inspired by the well-known fact that the Shapley value selects an element of the core of any convex game. In general, the Shapley value is the barycenter of the marginal contribution vectors. Now, convex games are characterized by the property that all marginal contribution vectors are elements of the core (see Shapley 1971; Ichiishi 1981).

M. Hoffmann (B) Institute of Mathematical Economics (IMW), University of Bielefeld, Bielefeld, Germany e-mail: [email protected] P. Sudhölter (B) Department of Business and Economics, University of Southern Denmark, Odense, Denmark e-mail: [email protected]

558

M. Hoffmann, P. Sudhölter

Within the present paper, we consider (two-sided) assignment games as introduced by Shapley and Shubik (1972). As glove games are assignment games, further conditions are needed in order to guarantee that the Shapley value is in the core of an assignment game. Two properties of a convex game are interesting: (1) the core is its unique von Neumann–Morgenstern stable set. (2) It is exact. Solymosi and Raghavan (2001) showed that every exact assignment game has a stable core. We prove that the Shapley value of an exact assignment game is in its core. Moreover, by means of an example, we show that the Shapley value may not be a member of the core of an assignment game with a stable core. For a discussion of the relation of some other solutions and the core of assignment games see Raghavan and Sudhölter (2005). We now briefly review the contents of the paper. Section 2 recalls the definitions of the Shapley value, of assignment games, of exact games, and of the core. In Sect. 3 the main theorem, saying that the Shapley value of an assignment game is in its core, is formulated. We explain why it is possible in our context to directly apply Theorem 2 of Solymosi and Raghavan (2001) that describes simple properties of a square matrix that are necessary and sufficient to generate an exact assignment game. We use this result to derive a useful further characterization of exactness. Moreover, we present the aforementioned example, show that exactness of an assignment game is not necessary for core membership of the Shapley value, and characterize the 2 × 2 matrices that define assignment games whose Shapley value is a core member. Section 4 is devoted to the proof of the main result. Finally, in Sect. 5 we present an example showing that the main theorem cannot be generalized to arbitrary exact games with large cores. 2 Notation and definitions A (cooperative TU) game is a pair (N, v) such that ∅ = N is finite and v : 2N → R, v(∅) = 0. Let (N, v) be a game and n = |N|. Then the core of (N, v), C(N, v), is given by C(N, v) = {x ∈ RN | x(N) = v(N)

and

x(S) ≥ v(S) ∀S ⊆ N},

N N where x(S) = i∈S xi (x(∅) = 0) for every S ∈ 2 and every x ∈ R . In order to recall the definition of the Shapley value (see Shapley 1953), for each s = 0, . . . , n − 1, define γ (s) = s!(n−s−1)! . Then the Shapley value of (N, v), n! φ(N, v), is given by φi (N, v) =

γ (s)(v(S ∪ {i}) − v(S))

for all i ∈ N,

(2.1)

S⊆N\{i}

where s = |S|. Note that

S⊆N\{i}

γ (s) = 1.

(2.2)

The Shapley value of exact assignment games

559

The game (N, v) is exact if for any S ⊆ N there exists x ∈ C(N, v) with v(S) = x(S). Exact games were introduced by Shapley (1971) (see also Schmeidler 1972). In order to recall the definition of an assignment game, we say that, for finite sets S and T, an assignment of (S, T) is a bijection b : S → T such that S ⊆ S, T ⊆ T, and |S | = |T | = min{|S|, |T|}. We shall identify b with {(i, b(i)) | i ∈ S }. Let B(S, T) denote the set of assignments. A game (N, v) is an assignment game if there exist a partition {P, Q} of N and a nonnegative real matrix A = (aij )i∈P,j∈Q such that aij for all S ⊆ N. (2.3) v(S) = max b∈B(S∩P,S∩Q)

(i,j)∈b

Conversely, if P and Q are disjoint finite nonempty sets and if A = (aij )i∈P,j∈Q is a nonnegative real matrix, then let (P ∪ Q, vA ) be the assignment game associated with A, that is, v = vA is defined by (2.3). We conclude this section by recalling a result of Shapley and Shubik (1972). Let N = P ∪ Q. Then C(N, vA ) = {x ∈ RN + | x(N) = vA (N)

and

xi + xj ≥ aij ∀ (i, j) ∈ P × Q}. (2.4)

3 Core membership of the Shapley value Our main result is the following theorem. Theorem 3.1 If (N, v) is an exact assignment game, then φ(N, v) ∈ C(N, v). We postpone the proof of the foregoing theorem to Sect. 4. This section serves to prepare the proof and to show that if exactness is relaxed, then the statement of our result is no longer valid. For assignment games, exactness is not a necessary condition for core membership of the Shapley value (see Example 3.4). However, if just four-person assignment games are considered, a simple necessary and sufficient condition on the assignment matrix exists (see Proposition 3.5). Let (N, v) be an exact assignment game and let P and Q be finite nonempty sets, and let A be a nonnegative real P × Q matrix such that N = P ∪ Q and v = vA . In order to prove that φ(N, v) ∈ C(N, v), we may always assume that A is a square matrix, that is, p = |P| = |Q|. Indeed, we may add zero columns or zero rows if necessary, because the Shapley value and the core both satisfy the strong null player property1 . Also, by anonymity of the Shapley value and the core, we shall assume that P = {1, . . . , p}, Q = {1 , . . . , p }, and v(N) = aii . (3.1) i∈P 1 A solution satisfies the null player property, if any element of the solution to a game assigns zero to any null player of the game. It satisfies the strong null player property if, additionally, the solution to a game (N ∪ {k}, w) that arises from (N, v) by adding the null player k, arises from the solution to (N, v) by adding a zero coordinate for the null player k to any element of the solution to (N, v).

560


If S ⊆ N, then we define S∗ = {i ∈ Q | i ∈ S} ∪ {i ∈ P | i ∈ S}.

(3.2)

So, S∗ is the set of all partners of players in S according to the optimal diagonal assignment of N. Note that |S∗ | = |S| and that (S∗ )∗ = S for all S ⊆ N. In order to prove the following useful lemma, we first recall the following necessary and sufficient condition for the exactness of an assignment game. The matrix A has a • dominant diagonal if aii ≥ aij and aii ≥ aji for all i, j ∈ P; • doubly dominant diagonal if aii + ajk ≥ aik + aji for all i, j, k ∈ P. Theorem 3.2 (Solymosi and Raghavan 2001) The assignment game associated with A is exact if and only if A has a dominant and a doubly dominant diagonal. Moreover, Solymosi and Raghavan (2001) show in their Theorem 1 that an assignment game associated with A has a stable core if and only if A has a dominant diagonal. The following example shows that Theorem 3.1 fails if the condition of exactness is replaced by core stability. Example 3.3 Let (N, v) be the assignment game associated with the matrix ⎞

⎛ 8

⎜ ⎜ A = ⎜4 ⎝ 8

4 4 1

8

⎟ ⎟ 1⎟. ⎠ 8

As A has a dominant diagonal, the core of (N, v) is stable. Also, (N, v) is not exact, because a11 + a23 = 8 + 1 < 8 + 4 = a13 + a21 . Applying (2.1) yields 1 (82, 39, 79, 82, 39, 79), where the first three components refer to the φ(N, v) = 20 1 “row players”. As 20 (39 + 39) < 4, φ(N, v) ∈ / C(N, v). Note that the foregoing example is minimal in the following sense. For every assignment game with a stable core and with less than six players, the Shapley value is in the core. Indeed, if there are five or less players, then we may assume that |P| ≤ |Q|, that is |P| ≤ 2. By adding zero rows, if necessary, and renaming the players in such a way that the diagonal is an optimal matching, we receive a square matrix with at most 2 nonzero rows. So the matrix may only have a dominant diagonal if it has at most 2 nonzero columns as well. However, in this case the diagonal is automatically doubly dominant, because every 2 × 2 matrix has a doubly dominant diagonal. The following example shows that there exists a non-exact transitive assignment game (a game whose symmetry group is transitive) with a stable core, whose Shapley value is in its core.


561

Example 3.4 Let ⎞

⎛ 2

⎜ ⎜ A = ⎜0 ⎝ 2

2 2 0

0

⎟ ⎟ 2⎟ ⎠ 2

and v = vA . Note that (N, v) is not exact, because a11 +a32 = 2 < 4 = a12 +a31 . Also, if π is the cyclic permutation defined by 1 → 1 → 3 → 3 → 2 → 2 → 1, then v(π(S)) = v(S) for all S ⊆ N. So, applying the k-fold composition π k , k = 0, . . . , 5, to N leaves the game unchanged and transitivity of its symmetry group is shown. As the Shapley value is anonymous and Pareto optimal, we conclude that φ(N, v) = (1, 1, 1, 1, 1, 1). By (2.4), φ(N, v) ∈ C(N, v). In order to present a characterization of the set of 2 × 2 matrices such that v = vA and φ(N, v) ∈ C(N, v), it is useful to check the following property of the Shapley value of assignment games. Let v = vA be an assignment game associated with the P × Q matrix A such that (3.1) is valid. Then for any t ≥ max(i,j )∈P×Q −aij we claim that, with A + t = (aij + t)i,j∈P , φ(N, vA+t ) =

t + φ(N, vA ). 2

(3.3)

To see this, let w = vA+t and observe that w(S) = v(S) + t min{|S ∩ P|, |S ∩ Q|} for any S ⊆ N. By p = q, we may conclude that, for each i ∈ N and S ⊆ N \ {i},

w(S ∪ {i}) − w(S) + w(N \ S) − w(N \ (S ∪ {i})) = t + v(S ∪ {i}) − v(S) + v(N \ S) − v(N \ (S ∪ {i})) .

As γ (|S|) = γ (|N| − |S| − 1), our claim follows directly from (2.1).

⎞

⎛ a

b

⎠ c d such that v = vA is a non-exact assignment game and x = φ(N, v) ∈ C(N, v). By (3.1), Theorem 3.2, and anonymity we may assume that Now we are able to characterize the set of 2 × 2 matrices A = ⎝

b>a

and

c < d.

(3.4)

As x ∈ C(N, v), (2.4) implies that x1 + x1 = a. By (2.1) we may conclude that c < a (otherwise x1 + x1 > a)2 . Further computations show that a = d. Finally, 2 Indeed, if c ≥ a, then by (3.3), y = x − a is the Shapley value of v A−a where A − a = 2 ⎛ ⎞

⎝

0

b−a

⎠. As b > a, (2.1) implies that y1 > 0. Also, y1 ≥ 0 so that x1 + x1 > a, which is c−a d−a excluded for core elements.

562


applying the inequality x1 + x2 ≥ b yields 4a ≥ 3b + c. In view of Theorem 3.1 we have derived the following result.

Proposition 3.5 Let p = 2 and let A = aij i,j∈P be nonnegative such that (3.1) is valid with v = vA . Put a = min{a11 , a22 }, b = max{a12 , a21 }, and c = min{a12 , a21 }. Then φ(N, v) ∈ C(N, v) ⇔ (a ≥ b or (a11 = a22 and 4a ≥ 3b + c)) . Now, we continue with our preparations of the proof of Theorem 3.1. Note that, if (N, vA ) is the assignment associated with the P × Q matrix A such that (3.1) is satisfied, then, by (2.3) and (3.1), vA (S ∩ S∗ ) =

aii

for all S ⊆ N.

(3.5)

i∈P∩S∩S∗

Lemma 3.6 The assignment game (N, v) associated with A is exact if and only if v(S) = v(S ∩ S∗ ) + v(S \ S∗ )

for all S ⊆ N.

(3.6)

Proof Assume that (3.6) is satisfied. Let i ∈ P and j, k ∈ P \ {i}. By (2.3), aii + ajk = v({i, j, i , k }) ≥ aik + aji , aii = v({i, j, i }) ≥ aji ,

and

aii = v({i, i , k }) ≥ aik .

Hence, A has a doubly dominant and a dominant diagonal. Now assume that A has a dominant and doubly dominant diagonal and let S ⊆ N and let T = S ∩ S∗ . We shall prove (3.6) by induction on |T| = t. Indeed, if t = 0, then T = ∅ and (3.6) is true. Assume now that t > 0. Let b be an optimal assignment for S, that is, b ∈ B(S ∩ P, S ∩ Q) and v(S) = (i,j )∈b aij . Let i ∈ T ∩ P. By the inductive hypothesis and (3.5) it remains to show that v(S) ≤ aii + v(S \ {i, i }). / b. If (i, i ) ∈ b, then the proof is complete. Hence, we may assume that (i, i ) ∈ The following three cases may occur. (1) There exists j ∈ S ∩ Q, j = i, such that (i, j ) ∈ b and i is not matched in b, that is, {(j, i ) | j ∈ P} ∩ b = ∅. As A has a dominant diagonal, v(S) = v(S \ {i, i , j }) + aij ≤ v(S \ {i, i , j }) + aii ≤ v(S \ {i, i }) + aii . (2) The case that there exists j ∈ S ∩ P, j = i, such that (j, i ) ∈ b and i is not matched in b, that is, {(i, j ) | j ∈ P} ∩ b = ∅, may be treated analogously to (1).


563

(3) If (i, k ), (j, i ) ∈ b for some j, k ∈ P \ {i}, then v(S) = v(S \ {i, i , j, k }) + aik + aji ≤ v(S \ {i, i , j, k }) + aii + ajk ≤ v(S \ {i, i }) + aii ,

because A has a doubly dominant diagonal. 4 Proof of Theorem 3.1

Let (N, v) be the assignment game associated with the P × Q matrix A such that (3.1) is satisfied. Let x = φ(N, v). By (2.1), x ≥ 0. In view of (2.4) and (3.1) it suffices to show that xi + xi = aii xi + xj ≥ aij

for all i ∈ P, for all i, j ∈ P.

(4.1) (4.2)

Note that (4.1) implies the well-known Pareto efficiency of x, that is, x(N) = v(N). Let i ∈ P. Step 1 We shall show that xi + xi = aii . Let n = |N| and note that, by the definition of γ (·), γ (s) = γ (n − s − 1)

for all

s = 0, . . . , n − 1.

(4.3)

By (4.3) and (2.1) it suffices to find a bijection f : 2N\{i} → 2N\{i } such that, for all S ⊆ N \ {i}, n − 1 = |S| + |f (S)|, aii = (v(S ∪ {i}) − v(S)) + (v(f (S) ∪ {i }) − v(f (S))).

(4.4) (4.5)

For each S ⊆ N \ {i} let f (S) = N \ (S∗ ∪ {i }) (see (3.2) for the definition of S∗ ). We shall now verify that f has the desired properties. Let S ⊆ N \ {i}. By / f (S) and definition i ∈ (4.6) f (S)∗ = N \ (S ∪ {i}). Hence, S = N \ (f (S)∗ ∪ {i}). So, f is a bijection with the desired range. As |S| = |S∗ | and i ∈ / S∗ , (4.4) is satisfied. In order to show (4.5), we distinguish two cases. / f (S)). By Lemma 3.6 and (3.5), (1) i ∈ S (that is, i ∈ v(S ∪ {i}) − v(S) = aii + v((S \ {i }) \ S∗ ) − v(S \ S∗ ), v(f (S) ∪ {i }) − v(f (S)) = v((f (S) ∪ {i }) \ f (S)∗ ) − v(f (S) \ f (S)∗ ).

564


Now, (4.5) follows, because, by (4.6), = (S \ {i }) \ S∗ , f (S) \ f (S)∗ = (S ∪ {i}) \ (S∗ ∪ {i }) i∈S∗

(f (S) ∪ {i }) \ f (S)∗ = (S ∪ {i}) \ S∗ = S \ S∗ . / S (that is, i ∈ f (S)) we may proceed analogously. Just (2) In the case that i ∈ the roles of S and f (S) and of i and i have to be exchanged. Step 2 Let j ∈ P. We shall now show that xi + xj ≥ aij . By the first step we may assume that j = i. By (2.2) and (4.3), 1=

γ (|S|) =

γ (|S|) + γ (|N \ (S ∪ {i})|) = 2

S⊆N\{i,i }

S⊆N\{i}

γ (|S|).

S⊆N\{i,i }

Let R = 2N\{i,i ,j,j } . By (4.3) applied to any R ∪ {i }, R ∈ R, and by (4.7), 1=2

γ (|R|) + γ (|R ∪ {i }|) + γ (|R ∪ {j}|) + γ (|R ∪ {j }|) R∈R

=

(4.7)

(4.8)

(2γ (|R|) + 6γ (|R| + 1)) .

R∈R

For R ⊆ N let r = |R|. By (4.8) it suffices to show that, for all R ∈ R, there are c(R) and d(R) such that xi + xj = R∈R (c(R)γ (r) + d(R)γ (r + 1)), c(R) ≥ 2aij and d(R) ≥ 6aij .

(4.9) (4.10)

By (2.1) and using s = |S|, xi + xj =

g(s) v(S ∪ {i}) + v(S ∪ {j }) − 2v(S)

S⊆N\{i,j }

+ γ (s + 1) 2v(S ∪ {i, j }) − v(S ∪ {j }) − v(S ∪ {i}) .

(4.11)

To every R ∈ R we assign three coalitions defined by fj (R) = (N \ R∗ ) \ {j},

fi (R) = (N \ R∗ ) \ {i },

fij (R) = (N \ R∗ ) \ {i, j }.

Note that fj (·), fi (·), and fij (·) are injective mappings on R. Observe that for any S ⊆ N \ {i, j } there exists a unique R ∈ R such that S coincides with one of the sets R, R ∪ {j}, R ∪ {i }, and fij (R). Similarly, note that there exists a unique R ∈ R such that S ∪ {i, j } coincides with one of the sets N \ R∗ , fj (R), fi (R), and


565

R ∪ {i, j }. Let R ∈ R and c(R) = v(R ∪ {i}) + v(R ∪ {j }) − 2v(R) + 2v(N \ R∗ ) − v((N \ R∗ ) \ {i}) − v((N \ R∗ ) \ {j }), let dj (R) = v(R ∪ {i, j}) + v(R ∪ {j, j }) − 2v(R ∪ {j}) + 2v(fj (R)) − v(fj (R) \ {i}) − v(fj (R) \ {j }), di (R) = v(R ∪ {i, i }) + v(R ∪ {i , j }) − 2v(R ∪ {i }) + 2v(fi (R)) − v(fi (R) \ {i}) − v(fi (R) \ {j }), dij (R) = v(fij (R) ∪ {i}) + v(fij (R) ∪ {j }) − 2v(fij (R)) + 2v(R ∪ {i, j }) − v(R ∪ {j }) − v(R ∪ {i}), and put d(R) = dj (R) + di (R) + dij (R). Thus, (4.9) is implied by (4.11) and (4.3). Let R = R \ R∗ . Lemma 3.6 together with (3.5) yield R ∪ {i}) − v( R ∪ {i }) − v( R ∪ {j}) + v( R ∪ {j }), c(R) = aii + ajj + v( R ∪ {j}) + 2v( R ∪ {j }) + v( R ∪ {i, j}) − v( R ∪ {i , j }), dj (R) = aii + ajj − 2v( R ∪ {i}) − 2v( R ∪ {i }) − v( R ∪ {i, j}) + v( R ∪ {i , j }), di (R) = aii + ajj + 2v( R ∪ {i}) + v( R ∪ {i }) + v( R ∪ {j}) − v( R ∪ {j }) dij (R) = aii + ajj − v( R ∪ {i , j}). + 2v( R ∪ {i, j }) − 2v( We conclude that R ∪ {i}) − v( R ∪ {i }) − v( R ∪ {j}) + v( R ∪ {j }) d(R) = 3aii + 3ajj + v( R ∪ {i , j}). + 2v( R ∪ {i, j }) − 2v( Therefore, in order to verify (4.10) it suffices to prove that R ∪ {j }) − v( R ∪ {i }), aij ≤ aii + v( aij ≤ ajj + v( R ∪ {i}) − v( R ∪ {j}), R ∪ {i, j }) − v( R ∪ {j, i }). 2aij ≤ aii + ajj + v(

(4.12) (4.13) (4.14)

In order to show (4.12) we distinguish two cases. If i is not matched in an optimal R ∪ {i }) = v( R), then the desired inequality assignment of R ∪ {i }, that is, if v( is immediately implied, because A has a dominant diagonal. If i is matched to R \ {k}), some k ∈ P ∩ R in an optimal assignment, that is, if v( R ∪ {i }) = aki + v( then v(R ∪ {j }) ≥ akj + v(R \ {k}) implies that aii + v( R ∪ {j }) − v( R ∪ {i }) ≥ aii + akj − aki

566


and, so (4.12) is valid, because A has a doubly dominant diagonal. In a completely analogous way we may show (4.13). In order to prove (4.14), put R ∪ {i, j }) − v( R ∪ {j, i }). β = aii + ajj + v( R ∪ {j, i } Let b ∈ B(P ∩ ( R ∪ {j}), Q ∩ ( R ∪ {i })) be an optimal assignment for (see (2.3)). Four cases may occur. R ∪ {j, i }) = v( R) + aji . As v( R ∪ {i, j }) ≥ v( R) + aij , we Case 1 (j, i ) ∈ b. So, v( may conclude that β ≥ aii + ajj + aij − aji ≥ 2aij , because A has a dominant diagonal. Case 2 There exists some ∈ P ∩ R such that (, i ) ∈ b and j is not matched in R \ {}) and v( R ∪ {i, j }) ≥ aj + v( R \ {}). So, b. Then v( R ∪ {j, i }) = ai + v( β ≥ aii + ajj + aj − ai ≥ ajj + aij + ai − ai ≥ 2aij , where the second inequality is true, because A has a doubly dominant diagonal, and the third inequality holds, because A has a dominant diagonal. R such that (j, k ) ∈ b and i is not matched in Case 3 There exists k ∈ Q ∩ R \ {k }), v( R ∪ {i, j }) ≥ aik + v( R \ {k }) and, b. Then v( R ∪ {j, i }) = ajk + v( analogously to Case 2, β ≥ aii + ajj + aik − ajk ≥ aii + ajk + aij − ajk ≥ 2aij . R and ∈ P ∩ R such that (j, k ), (, i ) ∈ b. Then Case 4 There exist k ∈ Q ∩ v( R∪{j, i }) = ajk +ai +v( R\{, k }) and v( R∪{i, j }) ≥ aik +aj +v( R\{, k }). Applying the fact that A has a doubly dominant diagonal twice yields, β ≥ aii + ajj + aik + aj − ajk − ai ≥ aij + ai + ajk + aij − ajk − ai = 2aij . Remark 4.1 (1) The foregoing proof does not use Iñarra and Usategui (1993) who presented a condition that is equivalent to φ(N, v) ∈ C(N, v) in their Theorem 1. In order to verify this condition for a general TU game, 2|N|−1 inequalities have to be checked. Hence, Proposition 3.5 significantly simplifies the foregoing condition for 2 × 2 assignment games. (2) It can easily be verified that an exact assignment game (N, v) is a partially average convex game as introduced in the aforementioned paper if and only if the underlying matrix is a diagonal matrix, that is, (N, v) is convex, provided that p ≥ 3. 5 An exact game with a large core In this section, we shall present an exact TU game (N, v) whose core (a) does not contain the Shapley value and (b) is large, that is, if y ∈ RN satisfies y(S) ≥ v(S)


567

for all S ⊆ N, then there exists x ∈ C(N, v) such that x ≤ y. (Sharkey (1982) introduced largeness of the core and showed that this property implies core stability.) Thus, this example complements Theorem 3.1: Indeed, according to Solymosi and Raghavan (2001), an assignment game is exact if and only if it has a large core. Let N = {1, . . . , 5}, let λ1 , λ2 , λ3 ∈ RN be given by λ1 = (1, 1, 1, 0, 0),

λ2 = (1, 1, 0, 1, 0),

λ3 = (0, 0, 1, 1, 1),

and let v be defined by v(S) = min λr (S) r=1,2,3

for all S ⊆ N.

As λr (N) = 3 for all r = 1, 2, 3, the game (N, v) is exact. By (2.1), φ(N, v) = φ =

1 (36, 36, 41, 41, 26). 60

(In order to determine φ, just φ1 and φ5 have to be computed via (2.1), because players 1 and 2 are substitutes as well as 3 and 4 are.) Hence, v({2, 3, 4}) = 2 >

118 = φ({2, 3, 4}). 60

So, φ(N, v) ∈ / C(N, v). We are left to prove the following lemma. Lemma 5.1 The game (N, v) has a large core. Proof Let X = {x ∈ RN | x(S) ≥ v(S) for all S ⊆ N}. Taking into account that v is monotonic (v(S) ≤ v(T) for all S ⊆ T ⊆ N), that 1 and 2 as well as 3 and 4 are substitutes, and a careful inspection of the coalition function v yields that x ∈ X if and only if x ≥ 0;

(5.1)

x3 + x4 ≥ 1; xi + xj ≥ 1 for all i ∈ {1, 2} xi + x3 + x4 ≥ 2

for all i ∈ {1, 2}.

and j ∈ {3, 4, 5};

(5.2) (5.3) (5.4)

Note that (5.3) and (5.4) imply x(N) ≥ 3 = v(N). Let x ∈ X and Y = {y ∈ X | y ≤ x}. Then Y is a nonempty compact (convex polyhedral) set. As y → y(N), y ∈ Y, is continuous, there exists z ∈ Y such that z(N) ≤ y(N) for all y ∈ Y. It remains to show that z(N) = v(N) = 3. By (5.1) one of the following 4 cases occurs. Case 1 z1 = 0. Then, by (5.3), zi ≥ 1 for all i = 3, 4, 5, and, by (5.1), z ≥ λ3 . As λ3 ∈ X, z = λ3 and the proof is complete.

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Case 2 The case z2 = 0 may be treated analogously. Case 3 z5 = 0. Then (5.3) implies z1 , z2 ≥ 1 and (5.2) implies that z3 + z4 ≥ 1. As z(N) is minimal, z ≥ 0 implies that z3 + z4 = 1 = z1 = z2 . Hence, z(N) = 3. Case 4 z1 , z2 , z5 > 0. Then z1 = z2 ≤ 2 by minimality of z(N). In view of (5.2), z1 = z2 ≤ 1. Again minimality implies that z1 + z5 = z2 + z5 = 1. By (5.4), z3 + z4 ≥ 2 − z1 . So it remains to show that z3 + z4 = 2 − z1 . Assume the contrary. As 3 and 4 are substitutes, we may assume that z3 ≥ z4 . Then z3 > 1 − z21 . As z1 ≤ 2, there exists > 0 such that z3 − ≥ 0, z1 + z3 − ≥ 1 and z3 + z4 − ≥ 2 − z1 and a contradiction to the minimality of z(N) has been obtained. Acknowledgements We are grateful to Tamas Solymosi for helpful discussions and to the Editor and an anonymous referee of this journal for their remarks that helped to improve the writing of this paper. M. Hoffmann is grateful for financial support from the German Science Foundation (DFG). P. Sudhölter is grateful for financial support from the Spanish Ministry of Education and Science, Project No. SEJ2004-00968.

References Ichiishi T (1981) Super-modularity: applications to convex games and to the greedy algorithm for LP. J Econ Theory 25:283–286 Iñarra E, Usategui JM (1993) The Shapley value and average convex games. Int J Game Theory 22:13–29 Raghavan TES, Sudhölter P (2005) The modiclus and core stability. Int J Game Theory 33:467–478 Schmeidler D (1972) Cores of exact games. J Math Anal Appl 40:214–225 Shapley LS (1953) A value for n-person games. In: Contribution to the theory of games II, vol 28, Annals of Mathematics Studies, pp. 307–317, Princeton University Press, Princeton Shapley LS (1971) Cores of convex games. Int J Game Theory 1:11–26 Shapley LS, Shubik M (1969) Pure competition, coalitional power, and fair division. Int Econ Rev 10:337–362 Shapley LS, Shubik M (1972) The assignment game I: The core. Int J Game Theory 2:111–130 Sharkey WW (1982) Cooperative games with large core. Int J Game Theory 11:175–182 Solymosi T, Raghavan TES (2001) Assignment games with stable core. Int J Game Theory 30: 177–185